Most ar barrels will handle both rounds well. For peace of mind a barrel labeled 5.56 will safely handle both rounds and the supposed loss of accuracy potential is minimal. Check out Wilson Combat barrels.
We need a spotlight signal for Catshooter like Batman had
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To the OP, grab a cold beverage, type 223 vs 5.56 in the white bar labeled "Official Predator Masters Search Engine" at the top of the page and replenish beverages as needed. You'll be there a while